Originally published as JCO Early Release 10.1200/JCO.2006.05.5988 on August 22 2006

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Allometric Scaling Law Questions the Traditional Mechanical Model for Axillary Lymph Node Involvement in Breast Cancer

Romano Demicheli, Elia Biganzoli, Patrizia Boracchi, Marco Greco, William J.M. Hrushesky, Michael W. Retsky

From the Departments of Medical Oncology, Medical Statistics and Biometry, and Breast Surgery, Istituto Nazionale Tumori; Medical Statistics and Biometry, Università di Milano, Milano, Italy; The University of South Carolina, Dorn Veteran's Administration Medical Center, Columbia, SC; and the Department of Vascular Biology, Children's Hospital and Harvard Medical School, Boston, MA

Address reprint requests to Romano Demicheli, MD, PhD, Istituto Nazionale Tumori di Milano, Via Venezian 1, 20137 Milano, Italy; e-mail: demicheli{at}istitutotumori.mi.it

	ABSTRACT

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
PURPOSE: To find a quantitative relationship between tumor size and frequency of axillary lymph node involvement.

PATIENTS AND METHODS: The frequency of axillary node involvement versus primary tumor volume was analyzed in 10 selected series of patients incorporating a total of 57,244 women with resectable breast cancer. The average number of events per unit volume resulting in tumor spread to axillary lymph nodes before tumor surgical removal (V)/V, was estimated under simple probabilistic assumptions.

RESULTS: The allometric scaling law (V)/V = 0.0586V^–0.7457 was estimated on the data, fitting the proportion of lymph node involvement on tumor volume V (in microliters). The estimate 0.7457 (95% CI, 0.7200 to 0.7713) suggests that the true scaling exponent, under the assumed model, may be the fractional value, which characterizes scaling relationships for a wide variety of biologic variables at both the whole organism level and organ level.

CONCLUSION: Results suggest that the phenomenon should be related to some internal structural trait of the tumor. The vascular network seems to be the best candidate. This result does not support a mere mechanical model of lymphatic tumor spread. A more complex biology-based model of lymphohematogenous spread is suggested, in which the axillary nodes draining the lymph from the primary tumor may become activated by factors produced by both tumor cells and tumor stroma, thus favoring cell-selective homing of otherwise circulating tumor cells. The success of fractal features related to the internal architecture brings additional support to the consideration of primary breast cancer as an organ-like structure.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
Living organisms display a remarkable variety of structures and sizes. Despite this incredible diversity and complexity, many of the most fundamental biologic processes manifest an extraordinary simplicity when viewed as a function of size, by allometric scaling laws that describe how biologic parameters vary with scale, regardless of the dissimilarities of the organisms being considered. The most famous allometric scaling law dates back to 1932, when an article was published¹ showing that the standard metabolic rates among mammals varied with the power of body mass. In 1934, the so-called elephant-to-mouse curve was confirmed.² Scaling laws arise from common underlying mechanisms that are independent of the specific nature of individual organisms. In particular, hierarchical fractal-like branching networks, which distribute energy and materials, are considered to play a central role.³

Breast cancer heterogeneity has been well known for many years. A variety of biologic variables such as grade, hormone receptors, HER2-neu status, and gene expression patterns may combine in different fashions covering a wide range of clinical features. Moreover, accumulating molecular data advocate the stratification of breast tumors into several groups and suggest the hypothesis of the occurrence of many different breast cancers.⁴ Despite this heterogeneity, it would not be surprising that some features may scale basically with tumor size.

Early quantitative studies about the relation between tumor size and axillary lymph node involvement simply stated that "the larger the tumor, the more likely axillary nodes will be positive."⁵ More recently some authors suggested a linear relation between tumor diameter (D) and frequency of patients with nodal involvement (node positive) for tumors with D = 1 to 5 cm.^6,7 A linear relation, however, fails to explain nodal status for both small and large D values. Indeed, even among small tumors (D < 0.5 to 1 cm) a significant percentage (20% to 25%) are node positive, whereas, conversely, approximately 30% of patients with larger tumors (D > 5 to 6 cm) have no nodal involvement (node negative).^5,8 Multivariate analyses about the relative prognostic value of the two factors failed to resolve the question, given that they were reported to behave independently at times and to show variable interdependence at other times.

It should be emphasized that no study has been performed aimed explicitly to ascertain quantitatively the specific mathematical formulation linking tumor size and axillary nodal involvement. All reported investigations were ancillary analyses of data sets collected for different aims (eg, clinical trials, epidemiologic surveys). For clinical purposes, the D value does not need to be measured with high accuracy when it is within coded ranges (eg, T1c = 1 < D 2 cm); moreover, axillary status may be assessed with a variety of modalities ranging from the systematic complete lymph node dissection to the removal of a node sample.

All of these points were considered in the present investigation that reports an analysis of 10 selected series of patients (nine published reports and a series from the Milan National Cancer Institute [Milan, Italy]) incorporating a total of 57,244 women with resectable breast cancer. Both this large number of patients and a tailored data analysis allowed us to find a relation between frequency of nodal involvement and tumor size.

	PATIENTS AND METHODS

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
The Milan data were obtained from three randomized clinical trials on conservative surgery for which patients were selected on the basis of tumor size (D < 2.5 cm). A search among published reports addressing the subject was performed and the following selection criteria were adopted: data of at least 1,000 patients with operable breast cancer had to be reported; both D and nodal involvement had to be acquired by pathologic assessment; the number of patients with a given tumor size had to be reported explicitly or at least quantifiable reliably from other reported data.

Table 1 shows the main characteristics of the selected reports.^5-13 The adopted criteria excluded some reports from the analysis; among them was the frequently mentioned article of Koscielny and Tubiana,¹⁴ the data of which did not meet the third criterion. The report of Rivadeneira et al¹¹ was used despite the number of patients (919) because it supplies node-positive frequency data corresponding to D values millimeter by millimeter for tumor sizes less than 1 cm, and, together with Milan data, allowed us to better assess their distribution in this critical range.

To address these problems, a specific data processing method was adopted. For all studies reporting the frequency for given D grouping without other information about the D value corresponding to the reported frequency, the median tumor size (D_m) for each group was selected as follows. For groupings with D 2 cm, D_m was selected following the D distribution of Milan and Rivadeneira et al¹¹ data. The D distribution pattern between 1 and 2 cm was considered valid between 2 and 3 cm. For diameters greater than 3 cm, D_m was further reduced by 1 mm in comparison with the values used in the 1- to 3-cm range, to take into some account the decreasing trend.

To provide a general description of the process, a mathematical formulation, which does not need any a priori assumption about the biologic mechanism underlying nodal involvement or about the growth modality of primary breast cancer, was considered convenient (details are listed in the Appendix [online only]). The events resulting in tumor spread to an axillary lymph node before tumor surgical removal were assumed to follow a nonhomogeneous Poisson process with an intensity function expressed by the Weibull model. The probability of no events resulting in tumor spread to an axillary lymph node is then expressed as

(1)

where N is the number of events, V is the volume of the spatial region occupied by a given primary, and ß are two parameters.

Equation 1 allows us to obtain an estimate of the parameters and ß from clinical data, and then to assess the average number of events per unit volume as a function of the tumor volume

(2)

Equation 2 describes the dynamics of occurrence of the events resulting in tumor cell homing within axillary nodes as a function of the time-dependent geometric tumor volume.

From equation 1, the expected relative frequency of node-negative cases F(N–) among patients bearing tumors of the same size is

(3)

According to equation 3, the effect of tumor volume V on the probability of axillary lymph nodes involvement, P(N+), was modeled by means of a generalized linear model assuming binomial error.¹⁵ The binomial generalized linear model corresponding to equations 1 and 3 has the so-called complementary log-log link, leading to the following linear form in the parameters

(4)

Empirical relative frequencies of axillary lymph node involvement F(N+) to be modeled were computed for each single study. Given that the evaluation was performed by collecting patients included in the same studies, a correlation in the outcome is expected, leading to the inflation of the model residual deviance (overdispersion). This leads to a bias in statistical tests and to the underestimation of the SE for model effects. Therefore, the procedures proposed by Williams¹⁶ was applied to achieve corrected estimates. Statistical analysis was performed with S-Plus 2000 software (Statistical Sciences, Seattle, WA).

	RESULTS

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
The pattern of raw data of the Milan series (Fig 1A) shows the effect due to pathologist rounding off, resulting in a redistribution of diameters with peaks placed at values ending with 0 and 5. In Figure 1A an empirical smoothing (by moving average) of the underlying pattern is also drawn, showing that a steep relative frequency growth for diameters less than 1 cm is followed by a near-constant pattern between 1 and 2 cm. This trend is roughly maintained up to 3 cm, whereas the curve drops regularly for larger diameters (Fig 1B).

View larger version (15K): [in this window] [in a new window]   Fig 1. (A) Tumor diameter distribution of the Milan series. Raw data show the effect due to the rounding-off. A smoothed evaluation (by moving average) of the underlying pattern is also drawn. (B) Tumor diameter distribution from Carter et al.7 After early growth, a near-constant pattern is maintained up to 3 cm, which decreases regularly with larger size.

View larger version (6K): [in this window] [in a new window]   Fig 2. Relative frequencies of node-positive patients for all series reported in Table 1. The curve represents the estimated relative frequency of node-positive patients as calculated from equation F(N+) = 1 – exp(–0.0586V0.254) (see text).

(5)

The 95% CI for the exponent is 0.7200 to 0.7713. The curve describing the estimated relative frequency of node-positive patients fits well with the experimental data (Fig 2).

	DISCUSSION

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
The correlation between the average number of events per unit volume resulting in axillary node involvement and the tumor size, which describes the dynamics of the metastatic process during tumor growth, unexpectedly, is a typical allometric scaling law. Moreover, remarkably, the scaling exponent 0.7457 (95% CI, 0.7200 to 0.7713) approximates the value that is found in a wide variety of biologic phenomena.

Allometric scaling phenomena have been investigated widely and the dependence of several biologic variables Y on body mass M proved to be characterized by the scaling law Y = constant M^b, where b is almost always a multiple of . For example, diameters of aortas scale as M^3/8, heartbeat frequency scales as M^–1/4, and blood circulation time scales as M^1/4.¹ Several explanatory models have been proposed for the allometric scaling with body mass (reviewed by Agutter et al¹⁷). The most persuasive model, recently proposed by West et al,^3,18 addresses the supply of materials (eg, oxygen) to cells through hierarchical networks of branching tubes (eg, the circulatory system). The power is shown to be inherent in the geometry of branching networks, the crucial feature of which is the size invariance of the terminal units (capillaries in the circulatory system). According to this model, the total number of capillaries scales as M^3/4 rather than as M, and the volume serviced by each capillary scales as M^1/4. Although this model triggered a few critical remarks when extended to all biologic levels (reviewed by Agutter and Wheatley¹⁷), its validity for mammals is widely accepted. The allometric scaling law has even been used at the organ level to describe, for example, the correlation of individual fetal organ/tissue weights with the total fetal weight.¹⁹ In this context, the tissue density may be considered constant, and weight and volume exhibit direct proportionality.

The result of our study suggests that axillary node involvement should be related to some trait of the tumor that scales as M^3/4. By meta-analysis, evidence was provided that "there is, in general, a lack of correlation between tumor size and various biologic prognostic factors"²⁰ and this result was confirmed by clustering analysis.²¹ In particular, results of studies on estrogen receptor expression versus tumor size have been inconsistent, and most of them, including prospective studies,²² failed to show evidence of correlation.²³ In addition, HER2 overexpression was found to be uncorrelated to tumor size.²⁴ Histologic grading was reported to have a trend to a worse score for larger tumors²⁰ and also to be uncorrelated with tumor size.²¹ Therefore, all major known biologic factors underlying breast cancer heterogeneity reasonably can be considered as uninvolved, or marginally involved at the most, in the scaling phenomenon, even if they might be implicated in determining the baseline biologic risk of the axillary node involvement. The issue of whether the scaling phenomenon changes across biologic factors should be addressed by targeted studies.

Given that other known tumor traits may be reasonably considered irrelevant, we suggest that, in analogy with the concepts underlying the West model³, the vascular network may be the most reasonable involved tumor structure. The hypothesis may fit well for hematogenous metastases, given that endothelial surface is the actual surface separating the tumor compartment from the whole body compartment. Yet, it conflicts with the widely accepted model of mechanical tumor spread from the primary to axillary lymph nodes via lymphatic vessels. Therefore, how can the node involvement process suggested by the West model be reconciled with established data about tumor spread to regional lymph nodes? First, is inner tumor configuration compatible with the West model main assumptions?

Assuming the West model for allometric scaling implies the assumption of the size invariance of the capillaries, which results in the scaling of capillary density per cross-sectional area of the tissue as M^–1/12.¹⁸ Unfortunately, a significant correlation between microvessel density (MVD) and tumor size, which would strongly support the appropriateness of the model, has not been reported. Following the early report from Weidner et al,²⁵ angiogenesis has been studied widely, essentially by assessing, in histologic sections, the MVD in the so-called hot spots; most reports failed to find associations between MVD and tumor size. The MVD assessment technique, however, displayed a wide variability within and between reports; therefore, data accuracy might have been inadequate to detect the fairly weak size dependence of the capillary density that would decrease only approximately 20% when tumor diameter doubles.

Although direct proof of its validity is lacking, some reports provide information about the tumor structure, supporting the plausibility of adopting the West model. Microvessel density may decrease progressively from peripheral to inner areas where the vascular architecture is impaired to a varying extent, with irregular perfusion.²⁶ However, although the conditions resulting from the vascular and perfusional impairment may induce focal necrosis and stromal changes to a variable degree, the interior of the tumor shows viable and vascularized tissue, with both tumor and stroma cells. In particular, for breast cancer, the inner areas showed vessel density similar to normal levels,²⁶ and it seems reasonable to believe that the main assumptions of the model, in general, may be valid.

The mechanical model of lymphatic tumor spread (primary tumor lymphatic capillaries collecting lymphatic trunks axillary lymph nodes) has been the conceptual frame for radical axillary lymph node dissection and, more recently, for sentinel node biopsy. However, the lack of direct identification, until recent times, of lymphatic vascular endothelium prevented the support of the model with detailed findings. Even if the scenario is changing after the discovery of some factors and markers (vascular endothelial growth factor C, podoplanin, homeobox prospero-like protein 1, and lymphatic vessel endothelial receptor 1), lymphatic spread is far from clarified (reviewed by Pepper et al²⁷). Rather, a number of findings seem to contradict the mechanical model instead of supporting it. It has been reported in the past²⁸ and confirmed recently ²⁹ that breast tumors lack an intrinsic lymphatic network, and it has been proposed that functional lymphatics in peritumoral stroma, sometimes containing neoplastic cells, are responsible for lymphatic dissemination.³⁰ However, the findings that high MVD correlates with the occurrence of nodal metastases,³¹ and that the relative risk of axillary node invasion depends on proliferative activity of intratumoral fibroblasts³² emphasize the concept that inner tumor structures may have a more prominent role than peritumoral lymphatic vessels.

Before accepting the hypothesis that the hematogenous route might be the most important route to axillary node involvement, we need to explain the orderly invasion of nodes beginning with the sentinel node(s). Cells from the tissue surrounding cancer cells are not idle bystanders but rather active participants in the tumor development.^33,34 Cancer cells produce stroma-modulating factors, similar to the process of wound healing, that activate fibroblasts (in particular), leading to the secretion of additional growth factors, cytokines, and proteases acting on cancer cells and cooperating with the establishment of an activated stroma.^35,36 An activated stroma can provide fertile soil even for the malignant cell homing, as experimental animal models^37,38 and clinical reports³⁹ demonstrated. It is not unreasonable to assume, therefore, that the axillary (sentinel) nodes draining the lymph from the primary tumor in the breast may become activated early by factors produced by both tumor cells and tumor stroma, thus favoring selective homing of otherwise circulating tumor cells. This occurrence has been described recently in an animal model of squamous cell carcinoma.⁴⁰ Moreover, it has been reported that changes in some lymph node cell populations and nodal metastases may represent independent processes even in breast cancer patients.⁴¹ This mechanism may be added to the reported role of some chemokines and chemokine receptors (CXCL12/CXCr4, CCL21/CCR7) in determining the selective seeding of lymph nodes and other specific sites in animal models.^42,43 Furthermore, this mechanism substantiates the statement by Fisher,⁴⁴ that "tumor bearing nodes are ‘indicators', not 'instigators’ of metastatic disease," which conflicts with the Virchow-Halsted paradigm.

The presence in breast tumors of allometric scaling properties, which are related to specific fractal features of the internal architecture of the studied body, brings additional support to considering them organ-like structures. Tumor properties, therefore, would not result from neoplastic cells only, but would reflect integrated behaviors of different cell types, including endothelial vasculature, fibroblasts, and inflammatory and immune cells. This picture is in keeping with recent reports stressing the crucial role that tumor stroma plays both in carcinogenesis and in tumor development and clinical behavior.^33,35,45-48 In this conceptual framework, it is not surprising that a peculiar tumor trait may reveal allometric properties. The results of this study, therefore, support indirectly the recent trend to shift from the reductionist somatic mutation theory of cancer to a more complex theory focused on the disruption of intercellular signaling.

	Appendix

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
Let S be the spatial region with volume V occupied by a given primary, and let s represent a small portion of it. Let us assume that within each of the different volumes, v_i of the s_i portions, the event rate _i is constant, and that the occurrence of an event resulting in tumor spread to an axillary lymph node before tumor surgical removal in the volume v_i is independent from other similar events in the same and other volume v_j. Therefore, the number of events, N_i in the ith volume, has the Poisson distribution with mean _iv_i, and the probability that N_i assumes a given value n is

(1)

If _i = constant for all v_i, we have an homogeneous Poisson process with expected number of events V. Otherwise, under the hypothesis that v_i are enough small to be treated as infinitesimal, the expected number of events in the region S is given by a function

where (x) is the event rate density function within S; that is, (x) is the intensity function of a nonhomogeneous Poisson process. Hence, the general formula for the probability distribution of the events in S is

(2)

Let us assume that tumors have a spheroid growth, such that S may be considered a sphere B = B_R(V) of volume V, and radius r = R(V) = (3V/4)^1/3, and that (x)=(r), where r is the distance from the center of the spherical tumor. Hence we have

where the integration is over the interval [0, r = (3V/4)^1/3]. The volume V identifies the spherical region, so that the symbol for the expectation may change from the original (S) (ie, a function of the set S) to the numerical function (V), and also the other functions of the spherical region of volume V can be considered numerical functions of V.

Substituting (V) in equation 1, we obtain that the random number of events N(V) in B_R(V) has the probability distribution

(3)

The process intensity (r) may be expressed in terms of the spherical volume v(r)=(4/3)r³, hence, coming back to the original symbol, we write (r)=(v). Assuming now that (v) has the form suggested by a Weibull model

where is a scale parameter and ß is the shape parameter, and integrating over the whole spherical region, we may obtain

Therefore the nonhomogeneous Poisson process spherical model with a Weibull intensity function has the distribution

The probability of no events resulting in tumor spread to an axillary lymph node is then expressed as

(4)

Equation 4 allows us to obtain an estimate of the parameters and ß from clinical data by the corresponding generalized linear model with binomial error, and then to assess the cumulative event rate (ie, the average number of events per unit volume) as a function of the tumor volume

(5)

Equation 5 provides a useful approach to understand the phenomenon of nodal involvement because it describes the dynamics of the occurrence of the events resulting in tumor cell homing within axillary nodes as a function of the geometric tumor volume.

From equation 4, the expected relative frequency of node-negative cases F(N–) among patients bearing tumors of the same size is

(6)

where F(N+) is the relative frequency of node-positive cases. According to equation 6, the aim of the analysis was to study the relative frequency of node-negative cases F(N–) among patients bearing tumors of the same size. Therefore, the effect of tumor volume V on the probability of axillary lymph nodes involvement P(N+) was modeled by means of a generalized linear model assuming binomial error.¹⁵ The binomial generalized linear model corresponding to equations 4 and 6 has the so called complementary log-log link leading to the following linear form in the parameters

Empirical relative frequencies of axillary lymph nodes involvement F(N+) to be modeled were computed for each single study. Given that the evaluation was performed by collecting patients included in the same studies, a correlation in the outcome is expected, leading to the inflation of the model residual deviance (overdispersion). This leads to a bias in statistical tests and to the underestimation of the SE for model effects. Therefore, the procedures proposed by Williams¹⁶ was applied to achieve corrected estimates. Statistical analysis was performed with S-Plus 2000 software (Statistical Sciences, Seattle, WA).

	Authors' Disclosures of Potential Conflicts of Interest

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 
The authors indicated no potential conflicts of interest.

	Author Contributions

TOP ABSTRACT INTRODUCTION PATIENTS AND METHODS RESULTS DISCUSSION Appendix Authors' Disclosures of... Author Contributions REFERENCES

 

Conception and design: Romano Demicheli Provision of study materials or patients: Romano Demicheli, Marco Greco Collection and assembly of data: Romano Demicheli, Elia Biganzoli, Patrizia Boracchi Data analysis and interpretation: Romano Demicheli, Elia Biganzoli, Patrizia Boracchi, William J.M. Hrushesky, Michael W. Retsky Manuscript writing: Romano Demicheli, Elia Biganzoli, Patrizia Boracchi, William J.M. Hrushesky, Michael W. Retsky Final approval of manuscript: Romano Demicheli, Elia Biganzoli, Patrizia Boracchi, Marco Greco, William J.M. Hrushesky, Michael W. Retsky

 

	ACKNOWLEDGMENTS

 
We thank Livio Triolo of the Dipartimento di Matematica, Universita' di Roma Tor Vergata, for his critical reading of the manuscript and for his suggestions about the mathematical model.

	NOTES

 
published online ahead of print at www.jco.org on August 21, 2006.

Authors' disclosures of potential conflicts of interest and author contributions are found at the end of this article.

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Submitted January 10, 2006; accepted July 12, 2006.

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